Significant research-and-development effort is currently being applied to the mathematical and computational field of optimization. Optimization techniques are applied to a wide variety of everyday and complex theoretical problems in order to find solutions to those problems, including cost-effective and time-efficient control strategies for complex systems and organizations. For example, optimization techniques may be applied to schedule labor and equipment resources within a large manufacturing organization in order to most cost-effectively produce a given amount of manufactured product to meet a projected demand for the product at various points in time. As another example, optimization techniques may be applied to a large control problem, such as the control of traffic lights within a city, in order to allow for as large a volume of traffic to move through the city as possible without generating unreasonable delays at various intersections and for those traveling on various secondary and tertiary routes within the city. Many natural systems can be viewed as seeking relatively optimal solutions to complexly defined problems governed by physical characteristics and principles and constrained by a large number of physical constraints. For example, complex chemical reactions can be considered to be governed by rather simple, thermodynamic principles by which individual components, such as molecules, seek minimum energy states and maximum entropy, constrained by various ways in which molecules can reconfigure themselves and exchange energy amongst themselves according to quantum mechanics. Even more complex biological systems can also be considered to be governed by chemical and thermodynamic principles as well as by analogous principles involving information content and exchange, compartmentalization and modular design, and other considerations. Thus, optimization problems may encompass an extremely wide variety of mathematically and computationally expressed models of natural phenomena, design, organization, and control of complex systems, and organization, transmission, and processing of information.
Current approaches for finding near optimal and optimal solutions for mathematically modeled problems are limited. When the number of decision variables and constraints in such problems increases from the small number of variables and constraints normally employed in simple, textbook problems to the large number of variables and constraints normally encountered in real-world systems, the computational resources required for seeking near optimal and optimal solutions increase exponentially in most cases. Current techniques cannot be satisfactorily applied to any but the simplest types of problems. Many currently available techniques involve applying oppressive constraints to optimization models, such as requiring variables to be continuous and requiring the hyper-dimensional volume representing the set of possible solutions of optimizations problems, or problem domain, to be convex.
Researchers, developers, system modelers, and investigators of many different types of complex system behavior have therefore recognized the need for more generally applicable and efficient methods and systems for optimization.